Abstract

1. Introduction

The general theory of nondegenerate submanifolds of Riemannian or semi-Riemannian manifolds is one of the most important topics of differential geometry [1, 2]. But the theory of degenerate or lightlike submanifolds of semi-Riemannian manifolds is relatively new [3] and different due to the fact that their normal vector bundle intersects with the tangent bundle. Thus, the study of lightlike submanifolds becomes more difficult than the study of nondegenerate submanifolds and one cannot use, in the usual way, the classical theory of submanifold to define induced objects on a lightlike submanifold. The degenerate geometry rises within the semi-Riemannian context, due to the existence of causal character of geometrical objects: their spacelike, timelike, or lightlike nature implies the existence of three types of hypersurfaces and submanifolds.

A lightlike framed hypersurface of a Lorentz -manifold, with an induced metric connection, is a Killing horizon and a globally hyperbolic spacetime; a de Sitter spacetime can carry a framed structure [3, 4]. Moreover, the contact geometry has a significant use in differential equations, optics, and phase spaces of dynamical systems [5–7]. All these motivated us to study lightlike hypersurfaces of indefinite globally framed -manifold, in particular indefinite -manifolds.

2. Preliminaries

A manifold of dimension is called a globally framed -manifold (-manifold) if it is endowed with a nowhere vanishing -tensor field of constant rank, such that is parallelizable; that is, there exist global vector fields , , with their dual -forms , satisfying
If the metric is a semi-Riemannian metric with index , such that
for any , accordingly is either spacelike or timelike, then the -manifold is called an indefinite metric -manifold. Then clearly , for any . An indefinite metric -manifold is called an indefinite -manifold if it is normal that is, the tensor field vanishes, where is Nijenhuis tensor of and , for any , where , for any . The Levi-Civita connection of an indefinite -manifold satisfies
where and . Also
and is integrable flat distribution, since , for detail see [8].

We recall notations and fundamental equations for lightlike hypersurfaces, which are due to Duggal and Bejancu [3].

Let be a -dimensional semi-Riemannian manifold with index , and let be a hypersurface of , with . is a lightlike hypersurface of if is of constant rank and the normal bundle is a distribution of rank on . A complementary bundle of in is a rank nondegenerate distribution over . It is denoted by and known as a screen distribution. Therefore we have
where is a orthogonal complementary vector bundle of in . The following theorem has important roles in studying the geometry of lightlike hypersurface.

Theorem 1. Let be a lightlike hypersurface of a semi-Riemannian manifold . Then there exists a unique vector bundle of rank over , such that for any nonzero section of on a coordinate neighborhood , there exists a unique section of on satisfying

Hence, we have the following decompositions of :

Let be the Levi-Civita connection on with respect to . Then using the decompositions in (7), Gauss and Weingarten formulae are given as
for any and , where and belong to while and belong to . Here is a torsion free linear connection on , is a -valued symmetric bilinear form on and known as the second fundamental form. is a linear operator on and known as the shape operator of lightlike immersion and is a linear connection on .

Locally, for the pair , following the Duggal and Bejancu's notation [3], we recall local second fundamental form and -form as
Hence locally, (8) becomes

If denotes the projection morphism of on then from (5), we have
for any , , where and belong to while and belong to . Here and are called the second fundamental form and the shape operator of the screen distribution, respectively. It should be noted that the induced linear connection is not a metric connection as it satisfies
where is a differential form locally defined on by . By direct calculation, we have
for any , , and . According to Duggal and Bejancu's notation [3], locally we have know
Then (11) become
and also give

From (9), it is clear that
that is, the second fundamental form of a lightlike hypersurface is degenerate. Following the notations of Duggal and Bejancu [3], for the curvature tensor of , we have , for any .

3. Lightlike Hypersurface of Indefinite -Manifolds

Let be a lightlike hypersurface of an indefinite -manifold , and such that the characteristic vector fields , are tangent to . As the ambient manifold has an additional geometric structure , we must look for a particular screen distribution on . Since , for any therefore is a distribution on of rank such that . Moreover , for any , therefore is tangent to . Since ; that is, the component of with respect to vanishes; this implies . As and are null vector fields satisfying , therefore from (2), we deduce that and are null vector fields satisfying . Hence is a vector subbundle of of rank . It is known [9] that if structure vector fields , are tangent to then belong to . Therefore implies that there exists a nondegenerate distribution of rank on such that
therefore
Let , and let then clearly is invariant and is anti-invariant under , and we have

Consider the local lightlike vector fields as
Using the decompositions in (21), any can be written as
where and are the projection morphisms of into and , respectively, and where , and is a local -form on defined by . Therefore
Applying to (23) and then using (24), we obtain . By assuming for any , we obtain a tensor field of type on and given by
Applying to (25) and then using the definition of , we obtain
Moreover, since therefore
for any . Thus we have the following theorem analogous to a theorem proved in [10].

Theorem 2. Let be an indefinite -manifold and let be a lightlike hypersurface of ; then is also a -manifold.

Since therefore applying to (26), we get , this implies that is an structure of constant rank on . Now using (2) and (25), we obtain
for any , where is a form locally defined on by . Replace by in (28) and using (26) and (27), we derive

Let be a lightlike hypersurface of an indefinite -manifold ; then by using (4), (10), and (25), we obtain
for any then comparing the tangential and transversal components, we get
Using (15) and (31), we get , taking inner product with , we get , and then using (29), we obtain

Definition 3. A lightlike hypersurface of an indefinite -manifold is said to be totally contact umbilical lightlike hypersurface if the second fundamental form of satisfies
for any , where is a transversal vector field on ; that is, , where is a smooth function on .

Remark 4. We can also write (35) as
where
and using (32)
for any . If , that is, ; then lightlike hypersurface is said to be totally contact geodesic.

Let be the curvature tensor fields of then for an indefinite -space form , we have (see [8])
for any , where and . Using (39), we obtain

Also, for the pair on , from (3.8) of page no. 94 of [3], we have
and , where

Lemma 5. Let be a totally contact umbilical lightlike hypersurface of an indefinite -manifold ; then for any , one has

Proof. By virtue of (37) and (42), we obtain
Now using (12) in (45), we have
Again using (12), we have
Thus from (46) and (47), the first expression of the theorem follows. Next, using (38) and (42), we obtain
Using (31) and (32), we get
and from (2), (3), (10), (16), (17), (18), and (25), we obtain
Thus by substituting (49) and (50) in (48), we obtain (44).

Theorem 6. Let be an indefinite -space form and be a totally contact umbilical lightlike hypersurface of . Then , where and satisfies the following differential equations:
for any .

Proof. Let be a totally contact umbilical lightlike hypersurface of an indefinite -space form of constant -sectional curvature . Then using (35), we have . Substituting (29), (44) in (43), we obtain
for any . Put in (52); we get
Put in (53) and then using , , ; we obtain
since , we get . Moreover, by putting and in (52), we obtain
Finally, putting , , and in (52) with and using that is nondegenerate, we obtain
Putting in (56), we get
By taking , we obtain
Since therefore using (19), we can write
where is an orthogonal basis of , then using (58), we have
which leads to get from (56)

Now, assume that there exists a vector field on some neighborhood of such that at some point in the neighborhood. Then from (61), it is clear that all the vectors of the fiber are collinear with . This contradicts . This implies the result.